Let AX = B be a system of three linear equations in three variables. Then the system has

(A) a unique solution if —A— = 0

(B) a unique solution if —A— ̸= 0

(C) no solutions if —A— = 0 and (adj A) B ̸= 0

(D) infinitely many solutions if —A— = 0 and (adj A)B = 0

Choose the correct answer from the options given below:

The correct answer is (B), (C) and (D) only.

Explanation:
Consider a system of three linear equations in three variables written in matrix form as:
$$AX=B$$
where $A$ is the coefficient matrix of order $3\times 3$, $X$ is the column matrix of variables, and $B$ is the column matrix of constant terms. The determinant of $A$ is represented as $\left|A\right|$ (or $—A—$ in the question statement).

Let's analyze the conditions step-by-step:

1. Unique Solution Condition ($\left|A\right|\ne 0$):
If the determinant of the coefficient matrix $A$ is non-zero ($\left|A\right|\ne 0$), then $A$ is a non-singular matrix. Consequently, the inverse matrix $A^{-1}$ exists.
We can multiply both sides of the matrix equation by $A^{-1}$:
$$A^{-1}\left(AX\right)=A^{-1}B$$
Using the associative property of matrix multiplication:
$$\left(A^{-1}A\right)X=A^{-1}B$$
Since $A^{-1}A=I$ (where $I$ is the identity matrix):
$$IX=A^{-1}B$$
$$X=A^{-1}B$$
This provides a single, unique solution for the system of equations. Thus, statement (B) is correct, and statement (A) is incorrect.

2. Singular Matrix Conditions ($\left|A\right|=0$):
If the determinant of the coefficient matrix $A$ is zero ($\left|A\right|=0$), the matrix is singular and the inverse $A^{-1}$ does not exist. In this case, we determine the consistency of the system by calculating the product of the adjoint matrix of $A$ and the constant matrix $B$, represented as $\left(\text{adj}A\right)B$:

• No Solution: If $\left(\text{adj}A\right)B\ne 0$, the system of linear equations is inconsistent and has no solutions. Therefore, statement (C) is correct.
• Infinitely Many Solutions: If $\left(\text{adj}A\right)B=0$, the system is consistent and has infinitely many solutions. Therefore, statement (D) is correct.

Consequently, statements (B), (C), and (D) are all correct, making the choice (B), (C) and (D) only the correct answer.